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Equation 31 · Part 9 · How a Model Actually Gets Small Enough to Run on a Phone

Denominator: 12

E ⁣[(x−x^)2]≈s212\mathbb{E}\!\left[(x - \hat{x})^2\right] \approx \frac{s^2}{12}
1212

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

12 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

for a signed integer of b bits. An 8-bit integer has 2^8=256 representable levels; a 4-bit integer has 2^4=16 . That sixteen-fold reduction in available codes is the entire cost of quantization in one number. Treating the rounding error as approximately uniform over one quantization step s , its expected squared magnitude is the classical result E ⁣[(x−x^)2]≈s212\mathbb{E}\!\left[(x - \hat{x})^2\right] \approx \frac{s^2}{12}. so halving the number of bits, which roughly doubles s at fixed range, roughly quadruples the expected squared error per weight. That is why INT8 is usually described as close to free and INT4 is not: the error a network has to absorb does not grow gently as bits are removed, it grows quadratically in the step size, and every bit…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.